Final answer:
By using the given that KN is parallel to ML and KN is congruent to ML, we can prove that triangles KLN and MNL are congruent by the AAS congruence criterion.
Step-by-step explanation:
To prove that triangles KLN and MNL are congruent, we will use the fact that KN is parallel to ML and KN is congruent to ML, which were given in the problem. Given this information, we can state that angle KLN is congruent to angle MLN and angle NLK is congruent to angle NLM because when two parallel lines are cut by a transversal, alternate interior angles are congruent.
Now, we have two pairs of congruent angles. To prove the triangles are congruent using the Angle-Angle (AA) similarity postulate for triangles, we need to show that two angles of one triangle are congruent to two angles of another triangle. This will imply that the triangles are similar and since one side KN is given as congruent to side ML, the triangles are not only similar but congruent by the Angle-Angle-Side (AAS) congruence criterion.
Lastly, the side LN is common to both triangles KLN and MNL, which means it is congruent to itself by the Reflexive Property of Congruence. Thus, according to the AAS congruence theorem, we can conclude that triangle KLN is congruent to triangle MNL.
Therefore, we have completed the proof using the properties of parallel lines, the AA similarity postulate, the AAS congruence criterion, and the Reflexive Property of Congruence.